This is a common
debate mistake, especially between two or more people of different fields. Arguing at cross purposes means carrying on an argument about different things, whether that be over divergent definitions of the
key terms, divergent notions of the
topic, or divergent standards of the proper way to resolve an argument.
One example of arguing at cross purposes because of different frameworks for resolving the debate would be two people arguing over what the weather would be like the next day. One watches the weather forcast and asserts that there's a 70% likelihood it will rain tomorrow. The other consults, for some culture clash, the I Ching and counters that it will actually be a bright, sunny day tomorrow. They argue and argue without getting anywhere, because their frameworks are entirely different. On the one hand, you have a person who's been taught to believe that modern scientific methods such as radar, sonar, and climate models most accurately predict the weather. On the other, you have a person who trusts in spiritual revelations. Both are going to be dogmatic about their assertions, and the argument will probably get nowhere.
Another example of arguing at cross purposes would be a tiff over who's the better runner. One person finishes marathons in record time, the other is the state champion in sprinting. They could both reasonable assert in their own minds that they are the better runner because the two have not agreed on a definition for their key term. Does 'runner' mean, 'sprinter'? Long distance runner? Some combination of the two? Until the usage of language is resolved, they will be arguing at cross purposes. For even stronger, more passionate examples, look at the abortion debate. How do you definte fetus? Why is your definition more accurate than the other persons? Without addressing these concerns, any debate is going to quickly descend into idiocy, as most abortion debates do.
Making sure that you're not arguing at cross purposes with your opponent is going to make any debate go more smoothly and intelligently. It is also important when writing a persuasive essay, in order to keep the reader engaged and the argument lucid.